Nonlinear collective flow reveals the breakdown of quadrupole--hexadecapole scaling in heavy ion collisions

Technical Summary: Nonlinear Collective Flow and the Breakdown of Quadrupole–Hexadecapole Scaling

Problem Statement
The intrinsic hexadecapole deformation ($\beta_4$) of atomic nuclei remains one of the least constrained properties of heavy nuclei. While the quadrupole deformation ($\beta_2$) determines the overall elongation, $\beta_4$ controls finer modifications of the nuclear surface, distinguishing between "waisted" and "barrel-like" longitudinal profiles. A long-standing challenge is determining the sign of $\beta_4$ and assessing the validity of the approximate geometric scaling relation $\beta_4 \propto \beta_2^2$. If this scaling holds, nuclei with identical $\beta_2^2$ values would exhibit degenerate geometric responses, rendering the independent contribution of $\beta_4$ experimentally indistinguishable. Conventional low-energy probes (e.g., electromagnetic transitions, Coulomb excitation) suffer from significant model dependence and provide only indirect access to higher-order multipole moments.

Methodology
To address these limitations, the authors utilize ultra-relativistic heavy-ion collisions as a probe, where the initial nuclear geometry is imprinted onto the quark–gluon plasma (QGP) and converted into final-state anisotropic flow. The study focuses on ultra-central ($0-5\%$) $^{238}\text{U}+^{238}\text{U}$ collisions at $\sqrt{s_{NN}} = 193$ GeV, comparing them with $^{197}\text{Au}+^{197}\text{Au}$ collisions at $\sqrt{s_{NN}} = 200$ GeV.

The analysis employs event-by-event viscous hydrodynamic simulations using the iEBE–VISHNU hybrid framework, which combines Monte Carlo Glauber initial conditions, viscous hydrodynamic evolution ($\eta/s = 0.08$), and UrQMD hadronic transport. The intrinsic nuclear density is modeled using a deformed Woods–Saxon distribution, allowing for independent variation of $\beta_2$ ($\pm 0.28$) and $\beta_4$ (ranging from $-0.10$ to $0.10$).

The core analytical strategy involves decomposing the fourth-order ($V_4$) and sixth-order ($V_6$) flow harmonics into linear and nonlinear response components:
$$V_4 = V_4^L + \xi_{4,22}V_2^2$$
$$V_6 = V_6^L + \xi_{6,222}V_2^3 + \xi_{6,33}V_3^2 + \xi_{6,24}V_2V_4^L$$
where $V_n^L$ represents the linear response to initial eccentricity, and $\xi$ terms represent nonlinear mode coupling coefficients. The authors expand observables around the spherical limit to isolate terms dependent on odd powers of $\beta_4$ and mixed nonlinear couplings (e.g., $\beta_2^2\beta_4$), which break the degeneracy implied by the scaling relation.

Key Results

1. Fourth-Order Flow ($v_4$): The ratio of flow harmonics $R(v_4\{2\}^2)$ in U+U relative to Au+Au shows a pronounced splitting between positive and negative $\beta_4$. This sensitivity is predominantly driven by the linear response ($V_4^L$), which carries independent information about the intrinsic hexadecapole deformation. For $|\beta_4| = 0.10$, the difference between barrel and waisted configurations reaches approximately 25% ($\sim 5\sigma$ significance).
2. Sixth-Order Flow ($v_6$): In contrast, the sensitivity of $v_6$ to the sign of $\beta_4$ originates almost entirely from nonlinear mode coupling. The linear response of $v_6$ is nearly insensitive to the sign of $\beta_4$, but the nonlinear contribution (specifically the coupling involving $V_4$) generates a strong topology-dependent splitting. The ratio $R(v_6\{2\}^2)$ increases monotonically with $\beta_4$, showing a separation of nearly 40% ($\sim 2\sigma$) between configurations.
3. Nonlinear Response Coefficients: The nonlinear response coefficient $\xi_{6,222}$ (quantifying the coupling $V_2^3 \to V_6$) cleanly separates all four intrinsic nuclear topologies defined by the signs of $\beta_2$ and $\beta_4$. While $\xi_{4,22}$ primarily distinguishes waisted from barrel-like shapes, $\xi_{6,222}$ resolves the full set of topologies with a separation exceeding $3\sigma$ (reaching $\sim 5\sigma$ for $|\beta_4|=0.10$). This coefficient retains simultaneous sensitivity to both deformation parameters through mixed cubic combinations.

Significance and Claims
The paper establishes that higher-order collective flow provides a direct probe of nuclear multipole structure, specifically enabling the isolation of the intrinsic hexadecapole deformation from the dominant quadrupole background. The primary findings are:

• Breakdown of Scaling: The distinct topological signatures observed in $v_4$ and $v_6$ provide experimental evidence for the breakdown of the approximate scaling relation $\beta_4 \propto \beta_2^2$.
• Mechanism of Information Transfer: The QGP acts as an efficient messenger that not only preserves but amplifies subtle geometric information. The sign of $\beta_4$ survives the QGP evolution and is enhanced through nonlinear hydrodynamic response.
• Experimental Accessibility: The sign of $\beta_4$ is identified as a measurable signature of deviations from the quadrupole–hexadecapole correlation. The nonlinear response coefficient $\xi_{6,222}$ is highlighted as a clean observable to distinguish the four intrinsic nuclear topologies.

The authors conclude that this framework offers a method to test quadrupole–hexadecapole correlations in relativistic heavy-ion collisions, with potential applicability to future measurements at RHIC and LHC and extensions to other nuclei with predicted hexadecapole deformations (e.g., $^{154}\text{Sm}$, $^{150}\text{Nd}$, $^{168}\text{Er}$, $^{176}\text{Yb}$, $^{208}\text{Pb}$, $^{232}\text{Th}$).